Lattice Sites in Cubic Unit Cell

We have so far discussed the symmetry present in patterns, crystals, and
the physical laws. If symmetry is understood as something that limits the
number of possible forms of matter and there can be no existence beyond that
boundary defined by this symmetrical world, then it can be found that it is
not totally correct. Actually, there is almost no existence of a matter in perfect
symmetrical state, and as a consequence, the symmetry must be treated as
no more than ideal norm from which there is always deviation in reality. If
this deviation is called asymmetry, then the problem of symmetry–asymmetry
must be understood more closely and intimately. Symmetry and asymmetry
are two closely related phenomena that exist in nature, in substances, and even
in physical laws, and in fact they are so closely interlinked that they must be
viewed as two aspects of the same concept. If beautiful gems and crystals
found in nature are the representation of symmetrical world, water in its bulk
structural form shows total asymmetrical arrangement of its molecules.
Therefore, when the aim of the book is to discuss the patterns, crystals,
and the symmetry that is manifested by them, it is also necessary to discuss
the deviations from symmetry, that is, asymmetry to make it complete or
more comprehensive.
The symmetry so far discussed in the earlier chapters consists of several “operations,” which when done on the object, the object comes to a stage of self
coincidence and there is no difference between the stages before and after the
said operations are done. This is due to invariance of these two positions or
stages. Now all these symmetries can be regrouped in two broad categories,
that is, (1) local order or symmetry and (2) long range order or symmetry
depending on the extent of their validity (Fig. 10.1).
This violation of the symmetry on the grain boundary results in a different crystal stage of matter known as polycrystalline state. There is marked
difference between the physical properties of these two different stages and
many of them, which are characteristic of the respective crystal stage, are important for enhancing the utility of the material. It may be emphasized that
the difference between the single crystal and polycrystal state of matter is the
randomness of the orientation of any particular plane throughout the bulk.
Now this randomness of the orientation will increase if the grain sizes become
finer, and this will lead to more asymmetry in one hand and more homogeneity of the physical properties of the material on the other. But there lies
enough space in “No man’s Land” between these two states of matter. During
grain growth state of the heat treatment, the randomness of the orientation of
any particular crystal plane will decrease. This is more conveniently achieved
by some mechanical processes of deformation. Now, this decrease in the order of randomness of the arrangement of crystallographic planes results in a
shift from random orientation to orientation in some preferred direction of the
plane. As a result, the homogeneity of physical properties is hampered, giving
rise to some heterogeneity in one hand and introduction of some symmetry in
otherwise asymmetric stage of matter. This is some times a boon while fabricating some mechanical structure or materials. This phenomena popularly
known as “texture” is, however, beyond the scope of discussion of this book
and it will suffice if it is mentioned here that this state of material is also a
symmetry in the world of asymmetry.
Property Single crystalline stage Polycrystalline stage
Symmetry Perfect symmetry in ideal
crystals
The symmetry is maintained
within the region known as
grain and remains “almost”
same but not exactly same
within other neighboring grains
belonging to same structure or
phase, but is totally different if
the grains are of different structures or phases
Order of
arrangement
of constituents
It remains same both locally and also throughout
the bulk
It changes at the boundary between two grains
Physical properties
As the physical properties
are direction dependent, a
single crystal shows total
heterogeneity
A polycrystal in this respect
shows homogeneous physical
properties
When close-packed structures mainly like FCC and HCP and also other
structures are deformed, first thing that happens is the fragmentation of the
grains called domains and polycrystalline materials, then shows more homogeneity, and the lattice is strained. This strained lattice contains higher energy
and resists more the deformation and thus inducts hardening. There appear
some drastic changes in their diffraction patterns. The stacking arrangement
of their close-packed planes also changes and this result in a defected region
compared to surrounding and is known as stacking fault. The number of planes
required to bring back the sequence into original are the number of faulted
planes. Less the number of faulted planes for a type of deformation process,
more is the energy required. Materials having more of this energy known as
stacking fault energy do not usually get “work hardened” (Fig.10.2).
The displacement of 111 plane by the vector b = 1/2[¯ 10¯ 1] in one step,
that is, from one A site to next A site requires larger misfit energy and so
A ? B ? C ? A is preferred by the two partials b1 and b2, satisfying the
relation b = b1 + b2, which is
a/2 [¯ 10¯ 1] = a/6 [¯ 211] + a/6 [¯ 1¯ 12].
The above situation may be visualized as follows:
A B C A B C A B C
? ? ? ? ? ?
A B C B C A B C A
104 10 Asymmetry in Otherwise Symmetrical Matte
? Faulted zone having h c p stacking sequence B C B C. type.
Now this faulted region having different stacking sequence does not commensurate with the perfect stacked regions on both sides. This may be seen
as an asymmetry introduced in the symmetrical structure.
Introduction of symmetry in otherwise asymmetrical structure is also
found in “super lattices” discovered in 1923 in AuCu3 alloys and found later
to exist in a number of alloys below a temperature known as critical temperature and they are PtCu3, FeNi3, MnNi3, and (MnFe) Ni3 alloys. Ordinarily
an alloy of say A and B elements exists in solid solutions wherein the atoms of
A and B are arranged randomly in the interstitials. This is the state of affairs
in the alloys other than those mentioned above. In these alloys, the random
structures are available at an elevated temperature, and when they are cooled
down below a particular temperature called critical temperature, an ordered
state happens wherein a particular set of lattice sites are occupied periodically
by say A atoms and the other particular sites by B atoms. The solution is
then said to be ordered and the lattice thus constituted is known by super
lattice. This is a sort of disorder–order transformation and is manifested by an
extra reflection in X-ray diffraction pattern. This is an important phenomena
not only because of the fact that this ordered state exhibits different physical
and chemical properties, but it is also an example of asymmetry to symmetry
transformation. The long range order that exist in the super lattice of AuCu3
alloys can be explained as follows:
In AuCu3 alloys, the occupancy probability for a particular lattice site say
by Au atoms is 1/4, then for Cu atoms it will be 3/4 because of the composition,
and the unit cell for the disordered and ordered structures will look as given
in Fig. 10.3.
The view from any side surface of the lattice will demonstrate the super
lattice more explicitly.
It can be seen from the above figures that in perfectly ordered state the
gold atoms occupy the corner positions and the copper atoms the face-centered
positions, whereas in the disordered state there is no such regularity and
positions in the unit cell are randomly occupied. As both individual structures
are cubic and have almost same lattice parameters, there is only a very slight
change in lattice parameter in the ordered state and so there is practically
no change in the positions of the X diffraction lines. But the change in the
positions of the atoms cause change in the diffracted intensities. Let us see
how it changes.
In the disordered state the structure factor F can be calculated as follows:
fav. = (atomic fraction of Au) fAu+ (atomic fraction of Cu) fCu,
fav. = 1/4fAu + 3/3fCu.
The positions of the atoms in the unit cell are 000, 1/21/20, 1/201/2, and
01/21/2.
The structure factor F = 
n
fne2?i(hxn + kyn + lzn) as there are four
atoms in unit cell (n = 4).
F = fav 1 + e?i(h+k) + e?i(h+1) + e?i(k+1)
For hkl unmixed, F = 4fav. = (fAu + 3fCu) and for hkl mixed F = 0.
Therefore, the disordered alloy produces the diffraction pattern similar to
face-centered cubic structure.
In the ordered state, each unit cell now should contain one Au atom at
000 position and Cu atoms at 1/21/20, 1/201/2, and 01/21/2, and the structure
factor then stands out as
F = fAu + fCu e?i(h+k) + e?i(h+1) + e?i(k+1) .
F = (fAu + 3fCu) when hkl are unmixed, but when hkl are mixed then F
instead of being zero it is F = (fAu ? fCu). Therefore, so far as the diffraction
lines are concerned, there exists one extra line for the reflection hkl mixed (odd
and even) only when the structure is perfectly ordered, otherwise it remains as
zero. This extra line is the manifestation of ordered structure and is known as
“super lattice line” even though they are weaker than fundamental lines.
So, super lattice is a transformation of one “disordered” or asymmetric
state to one ordered or symmetric state of this alloy and the order–disorder
transformation around a temperature establish that the symmetry and asymmetry may be viewed as the two sides of the same coin.
10.2 A Symmetry in Asymmetry I: Quasi Crystalline
State of Matter
It has been discussed in earlier Chapters (Chaps.3–5) that a perfect crystalline
structure should possess a long range order comprising both translational
and rotational symmetries, which should be maintained in three dimensions.
However, crystalline order can also be maintained in some ways other than
translational symmetry and they are called “aperiodic crystals.” Now, three
alternatives to translational symmetry are known: incommensurately modulated crystals, incommensurate composite crystals, and quasi crystals. The
modulated structures are obtained from the structures having translational
symmetry by giving displacements of the atoms in the periodic structure by
equal amounts. Incommensurate composite structures are formed in layered
compounds by two interpenetrating periodic structures which are mutually
incommensurate. The discovery of quasi crystals has added up one more dimension to crystallography. Influenced by the discovery of a number of quasi
crystals or quasi periodic crystals, International Union of Crystallography has
redefined the term crystals to mean “any solid having an essentially discrete
diffraction diagram.” This broader definition leads to the understanding that
microscopic periodicity are sufficient but not necessarily the only condition
for crystallinity. A distinct property of quasi crystals that has been found
from the diffraction pattern is that it shows fivefold rotation and also other
crystallographic point symmetries.
We have seen in Chap.3 (Fig.10.3) that there cannot be any crystalline
substance with fivefold of symmetry as the motifs having that symmetry cannot make any compact structure, and same is true for sevenfold, eightfold,
or tenfold rotation symmetries. It was accepted in classical crystallography